Properties

Label 1950.2.f.e.649.1
Level $1950$
Weight $2$
Character 1950.649
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1950,2,Mod(649,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.649
Dual form 1950.2.f.e.649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000i q^{12} +(3.00000 - 2.00000i) q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000 q^{18} +1.00000i q^{19} -2.00000i q^{21} -4.00000i q^{23} +1.00000i q^{24} +(-3.00000 + 2.00000i) q^{26} +1.00000i q^{27} +2.00000 q^{28} -5.00000 q^{29} -1.00000 q^{32} +2.00000i q^{34} -1.00000 q^{36} +7.00000 q^{37} -1.00000i q^{38} +(-2.00000 - 3.00000i) q^{39} +5.00000i q^{41} +2.00000i q^{42} -4.00000i q^{43} +4.00000i q^{46} +7.00000 q^{47} -1.00000i q^{48} -3.00000 q^{49} -2.00000 q^{51} +(3.00000 - 2.00000i) q^{52} -9.00000i q^{53} -1.00000i q^{54} -2.00000 q^{56} +1.00000 q^{57} +5.00000 q^{58} -4.00000i q^{59} +12.0000 q^{61} -2.00000 q^{63} +1.00000 q^{64} +7.00000 q^{67} -2.00000i q^{68} -4.00000 q^{69} +5.00000i q^{71} +1.00000 q^{72} -4.00000 q^{73} -7.00000 q^{74} +1.00000i q^{76} +(2.00000 + 3.00000i) q^{78} -5.00000 q^{79} +1.00000 q^{81} -5.00000i q^{82} -4.00000 q^{83} -2.00000i q^{84} +4.00000i q^{86} +5.00000i q^{87} -14.0000i q^{89} +(6.00000 - 4.00000i) q^{91} -4.00000i q^{92} -7.00000 q^{94} +1.00000i q^{96} +2.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} - 2 q^{9} + 6 q^{13} - 4 q^{14} + 2 q^{16} + 2 q^{18} - 6 q^{26} + 4 q^{28} - 10 q^{29} - 2 q^{32} - 2 q^{36} + 14 q^{37} - 4 q^{39} + 14 q^{47} - 6 q^{49} - 4 q^{51} + 6 q^{52} - 4 q^{56} + 2 q^{57} + 10 q^{58} + 24 q^{61} - 4 q^{63} + 2 q^{64} + 14 q^{67} - 8 q^{69} + 2 q^{72} - 8 q^{73} - 14 q^{74} + 4 q^{78} - 10 q^{79} + 2 q^{81} - 8 q^{83} + 12 q^{91} - 14 q^{94} + 4 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000i 0.408248i
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 3.00000 2.00000i 0.832050 0.554700i
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000i 0.229416i 0.993399 + 0.114708i \(0.0365932\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 0 0
\(26\) −3.00000 + 2.00000i −0.588348 + 0.392232i
\(27\) 1.00000i 0.192450i
\(28\) 2.00000 0.377964
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000i 0.342997i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −2.00000 3.00000i −0.320256 0.480384i
\(40\) 0 0
\(41\) 5.00000i 0.780869i 0.920631 + 0.390434i \(0.127675\pi\)
−0.920631 + 0.390434i \(0.872325\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000i 0.589768i
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 3.00000 2.00000i 0.416025 0.277350i
\(53\) 9.00000i 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 1.00000 0.132453
\(58\) 5.00000 0.656532
\(59\) 4.00000i 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 5.00000i 0.593391i 0.954972 + 0.296695i \(0.0958846\pi\)
−0.954972 + 0.296695i \(0.904115\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) 1.00000i 0.114708i
\(77\) 0 0
\(78\) 2.00000 + 3.00000i 0.226455 + 0.339683i
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.00000i 0.552158i
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 2.00000i 0.218218i
\(85\) 0 0
\(86\) 4.00000i 0.431331i
\(87\) 5.00000i 0.536056i
\(88\) 0 0
\(89\) 14.0000i 1.48400i −0.670402 0.741999i \(-0.733878\pi\)
0.670402 0.741999i \(-0.266122\pi\)
\(90\) 0 0
\(91\) 6.00000 4.00000i 0.628971 0.419314i
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 1.00000i 0.102062i
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 2.00000 0.198030
\(103\) 4.00000i 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −3.00000 + 2.00000i −0.294174 + 0.196116i
\(105\) 0 0
\(106\) 9.00000i 0.874157i
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 1.00000i 0.0957826i 0.998853 + 0.0478913i \(0.0152501\pi\)
−0.998853 + 0.0478913i \(0.984750\pi\)
\(110\) 0 0
\(111\) 7.00000i 0.664411i
\(112\) 2.00000 0.188982
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) −3.00000 + 2.00000i −0.277350 + 0.184900i
\(118\) 4.00000i 0.368230i
\(119\) 4.00000i 0.366679i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) −12.0000 −1.08643
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 17.0000i 1.50851i −0.656584 0.754253i \(-0.727999\pi\)
0.656584 0.754253i \(-0.272001\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −13.0000 −1.13582 −0.567908 0.823092i \(-0.692247\pi\)
−0.567908 + 0.823092i \(0.692247\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) 2.00000i 0.171499i
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 4.00000 0.340503
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 7.00000i 0.589506i
\(142\) 5.00000i 0.419591i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 3.00000i 0.247436i
\(148\) 7.00000 0.575396
\(149\) 14.0000i 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 3.00000i −0.160128 0.240192i
\(157\) 22.0000i 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 5.00000 0.397779
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 8.00000i 0.630488i
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 5.00000i 0.390434i
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 17.0000 1.31550 0.657750 0.753237i \(-0.271508\pi\)
0.657750 + 0.753237i \(0.271508\pi\)
\(168\) 2.00000i 0.154303i
\(169\) 5.00000 12.0000i 0.384615 0.923077i
\(170\) 0 0
\(171\) 1.00000i 0.0764719i
\(172\) 4.00000i 0.304997i
\(173\) 1.00000i 0.0760286i 0.999277 + 0.0380143i \(0.0121032\pi\)
−0.999277 + 0.0380143i \(0.987897\pi\)
\(174\) 5.00000i 0.379049i
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 14.0000i 1.04934i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) −6.00000 + 4.00000i −0.444750 + 0.296500i
\(183\) 12.0000i 0.887066i
\(184\) 4.00000i 0.294884i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 7.00000 0.510527
\(189\) 2.00000i 0.145479i
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) −24.0000 −1.72756 −0.863779 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 15.0000 1.06332 0.531661 0.846957i \(-0.321568\pi\)
0.531661 + 0.846957i \(0.321568\pi\)
\(200\) 0 0
\(201\) 7.00000i 0.493742i
\(202\) −2.00000 −0.140720
\(203\) −10.0000 −0.701862
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 4.00000i 0.278693i
\(207\) 4.00000i 0.278019i
\(208\) 3.00000 2.00000i 0.208013 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 9.00000i 0.618123i
\(213\) 5.00000 0.342594
\(214\) 3.00000i 0.205076i
\(215\) 0 0
\(216\) 1.00000i 0.0680414i
\(217\) 0 0
\(218\) 1.00000i 0.0677285i
\(219\) 4.00000i 0.270295i
\(220\) 0 0
\(221\) −4.00000 6.00000i −0.269069 0.403604i
\(222\) 7.00000i 0.469809i
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 6.00000i 0.399114i
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 1.00000 0.0662266
\(229\) 21.0000i 1.38772i 0.720110 + 0.693860i \(0.244091\pi\)
−0.720110 + 0.693860i \(0.755909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.00000 0.328266
\(233\) 24.0000i 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 3.00000 2.00000i 0.196116 0.130744i
\(235\) 0 0
\(236\) 4.00000i 0.260378i
\(237\) 5.00000i 0.324785i
\(238\) 4.00000i 0.259281i
\(239\) 24.0000i 1.55243i −0.630468 0.776215i \(-0.717137\pi\)
0.630468 0.776215i \(-0.282863\pi\)
\(240\) 0 0
\(241\) 10.0000i 0.644157i −0.946713 0.322078i \(-0.895619\pi\)
0.946713 0.322078i \(-0.104381\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000i 0.0641500i
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) −5.00000 −0.318788
\(247\) 2.00000 + 3.00000i 0.127257 + 0.190885i
\(248\) 0 0
\(249\) 4.00000i 0.253490i
\(250\) 0 0
\(251\) −13.0000 −0.820553 −0.410276 0.911961i \(-0.634568\pi\)
−0.410276 + 0.911961i \(0.634568\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 17.0000i 1.06667i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000i 0.499026i 0.968371 + 0.249513i \(0.0802706\pi\)
−0.968371 + 0.249513i \(0.919729\pi\)
\(258\) 4.00000 0.249029
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 13.0000 0.803143
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.00000i 0.122628i
\(267\) −14.0000 −0.856786
\(268\) 7.00000 0.427593
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) 20.0000i 1.21491i 0.794353 + 0.607457i \(0.207810\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) 2.00000i 0.121268i
\(273\) −4.00000 6.00000i −0.242091 0.363137i
\(274\) 13.0000 0.785359
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 12.0000i 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.00000i 0.298275i 0.988816 + 0.149137i \(0.0476497\pi\)
−0.988816 + 0.149137i \(0.952350\pi\)
\(282\) 7.00000i 0.416844i
\(283\) 6.00000i 0.356663i 0.983970 + 0.178331i \(0.0570699\pi\)
−0.983970 + 0.178331i \(0.942930\pi\)
\(284\) 5.00000i 0.296695i
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0000i 0.590281i
\(288\) 1.00000 0.0589256
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) −4.00000 −0.234082
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 0 0
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) 14.0000i 0.810998i
\(299\) −8.00000 12.0000i −0.462652 0.693978i
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 0 0
\(303\) 2.00000i 0.114897i
\(304\) 1.00000i 0.0573539i
\(305\) 0 0
\(306\) 2.00000i 0.114332i
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 2.00000 + 3.00000i 0.113228 + 0.169842i
\(313\) 11.0000i 0.621757i 0.950450 + 0.310878i \(0.100623\pi\)
−0.950450 + 0.310878i \(0.899377\pi\)
\(314\) 22.0000i 1.24153i
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 9.00000 0.504695
\(319\) 0 0
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 8.00000i 0.445823i
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 1.00000 0.0553001
\(328\) 5.00000i 0.276079i
\(329\) 14.0000 0.771845
\(330\) 0 0
\(331\) 20.0000i 1.09930i 0.835395 + 0.549650i \(0.185239\pi\)
−0.835395 + 0.549650i \(0.814761\pi\)
\(332\) −4.00000 −0.219529
\(333\) −7.00000 −0.383598
\(334\) −17.0000 −0.930199
\(335\) 0 0
\(336\) 2.00000i 0.109109i
\(337\) 22.0000i 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) −5.00000 + 12.0000i −0.271964 + 0.652714i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000i 0.0540738i
\(343\) −20.0000 −1.07990
\(344\) 4.00000i 0.215666i
\(345\) 0 0
\(346\) 1.00000i 0.0537603i
\(347\) 33.0000i 1.77153i 0.464131 + 0.885766i \(0.346367\pi\)
−0.464131 + 0.885766i \(0.653633\pi\)
\(348\) 5.00000i 0.268028i
\(349\) 6.00000i 0.321173i 0.987022 + 0.160586i \(0.0513385\pi\)
−0.987022 + 0.160586i \(0.948662\pi\)
\(350\) 0 0
\(351\) 2.00000 + 3.00000i 0.106752 + 0.160128i
\(352\) 0 0
\(353\) 1.00000 0.0532246 0.0266123 0.999646i \(-0.491528\pi\)
0.0266123 + 0.999646i \(0.491528\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 14.0000i 0.741999i
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) 21.0000i 1.10834i 0.832404 + 0.554169i \(0.186964\pi\)
−0.832404 + 0.554169i \(0.813036\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 8.00000 0.420471
\(363\) 11.0000i 0.577350i
\(364\) 6.00000 4.00000i 0.314485 0.209657i
\(365\) 0 0
\(366\) 12.0000i 0.627250i
\(367\) 13.0000i 0.678594i 0.940679 + 0.339297i \(0.110189\pi\)
−0.940679 + 0.339297i \(0.889811\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 5.00000i 0.260290i
\(370\) 0 0
\(371\) 18.0000i 0.934513i
\(372\) 0 0
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −7.00000 −0.360997
\(377\) −15.0000 + 10.0000i −0.772539 + 0.515026i
\(378\) 2.00000i 0.102869i
\(379\) 16.0000i 0.821865i 0.911666 + 0.410932i \(0.134797\pi\)
−0.911666 + 0.410932i \(0.865203\pi\)
\(380\) 0 0
\(381\) −17.0000 −0.870936
\(382\) −12.0000 −0.613973
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 4.00000i 0.203331i
\(388\) 2.00000 0.101535
\(389\) 25.0000 1.26755 0.633775 0.773517i \(-0.281504\pi\)
0.633775 + 0.773517i \(0.281504\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 3.00000 0.151523
\(393\) 13.0000i 0.655763i
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −33.0000 −1.65622 −0.828111 0.560564i \(-0.810584\pi\)
−0.828111 + 0.560564i \(0.810584\pi\)
\(398\) −15.0000 −0.751882
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 10.0000i 0.499376i 0.968326 + 0.249688i \(0.0803281\pi\)
−0.968326 + 0.249688i \(0.919672\pi\)
\(402\) 7.00000i 0.349128i
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 10.0000 0.496292
\(407\) 0 0
\(408\) 2.00000 0.0990148
\(409\) 16.0000i 0.791149i 0.918434 + 0.395575i \(0.129455\pi\)
−0.918434 + 0.395575i \(0.870545\pi\)
\(410\) 0 0
\(411\) 13.0000i 0.641243i
\(412\) 4.00000i 0.197066i
\(413\) 8.00000i 0.393654i
\(414\) 4.00000i 0.196589i
\(415\) 0 0
\(416\) −3.00000 + 2.00000i −0.147087 + 0.0980581i
\(417\) 0 0
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) 30.0000i 1.46211i −0.682318 0.731055i \(-0.739028\pi\)
0.682318 0.731055i \(-0.260972\pi\)
\(422\) −12.0000 −0.584151
\(423\) −7.00000 −0.340352
\(424\) 9.00000i 0.437079i
\(425\) 0 0
\(426\) −5.00000 −0.242251
\(427\) 24.0000 1.16144
\(428\) 3.00000i 0.145010i
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000i 0.722525i 0.932464 + 0.361262i \(0.117654\pi\)
−0.932464 + 0.361262i \(0.882346\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 19.0000i 0.913082i −0.889702 0.456541i \(-0.849088\pi\)
0.889702 0.456541i \(-0.150912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.00000i 0.0478913i
\(437\) 4.00000 0.191346
\(438\) 4.00000i 0.191127i
\(439\) −5.00000 −0.238637 −0.119318 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 4.00000 + 6.00000i 0.190261 + 0.285391i
\(443\) 21.0000i 0.997740i 0.866677 + 0.498870i \(0.166252\pi\)
−0.866677 + 0.498870i \(0.833748\pi\)
\(444\) 7.00000i 0.332205i
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) −14.0000 −0.662177
\(448\) 2.00000 0.0944911
\(449\) 9.00000i 0.424736i −0.977190 0.212368i \(-0.931882\pi\)
0.977190 0.212368i \(-0.0681176\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 21.0000i 0.981266i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −34.0000 −1.58011 −0.790057 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 24.0000i 1.11178i
\(467\) 23.0000i 1.06431i 0.846646 + 0.532157i \(0.178618\pi\)
−0.846646 + 0.532157i \(0.821382\pi\)
\(468\) −3.00000 + 2.00000i −0.138675 + 0.0924500i
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 4.00000i 0.184115i
\(473\) 0 0
\(474\) 5.00000i 0.229658i
\(475\) 0 0
\(476\) 4.00000i 0.183340i
\(477\) 9.00000i 0.412082i
\(478\) 24.0000i 1.09773i
\(479\) 31.0000i 1.41643i 0.705999 + 0.708213i \(0.250498\pi\)
−0.705999 + 0.708213i \(0.749502\pi\)
\(480\) 0 0
\(481\) 21.0000 14.0000i 0.957518 0.638345i
\(482\) 10.0000i 0.455488i
\(483\) −8.00000 −0.364013
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000i 0.0453609i
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −12.0000 −0.543214
\(489\) 4.00000i 0.180886i
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 5.00000 0.225417
\(493\) 10.0000i 0.450377i
\(494\) −2.00000 3.00000i −0.0899843 0.134976i
\(495\) 0 0
\(496\) 0 0
\(497\) 10.0000i 0.448561i
\(498\) 4.00000i 0.179244i
\(499\) 31.0000i 1.38775i 0.720095 + 0.693875i \(0.244098\pi\)
−0.720095 + 0.693875i \(0.755902\pi\)
\(500\) 0 0
\(501\) 17.0000i 0.759504i
\(502\) 13.0000 0.580218
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 5.00000i −0.532939 0.222058i
\(508\) 17.0000i 0.754253i
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 8.00000i 0.352865i
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −14.0000 −0.615125
\(519\) 1.00000 0.0438951
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −5.00000 −0.218844
\(523\) 6.00000i 0.262362i 0.991358 + 0.131181i \(0.0418769\pi\)
−0.991358 + 0.131181i \(0.958123\pi\)
\(524\) −13.0000 −0.567908
\(525\) 0 0
\(526\) 6.00000i 0.261612i
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 4.00000i 0.173585i
\(532\) 2.00000i 0.0867110i
\(533\) 10.0000 + 15.0000i 0.433148 + 0.649722i
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) 0 0
\(538\) −15.0000 −0.646696
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000i 0.429934i 0.976621 + 0.214967i \(0.0689643\pi\)
−0.976621 + 0.214967i \(0.931036\pi\)
\(542\) 20.0000i 0.859074i
\(543\) 8.00000i 0.343313i
\(544\) 2.00000i 0.0857493i
\(545\) 0 0
\(546\) 4.00000 + 6.00000i 0.171184 + 0.256776i
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) −13.0000 −0.555332
\(549\) −12.0000 −0.512148
\(550\) 0 0
\(551\) 5.00000i 0.213007i
\(552\) 4.00000 0.170251
\(553\) −10.0000 −0.425243
\(554\) 12.0000i 0.509831i
\(555\) 0 0
\(556\) 0 0
\(557\) 32.0000 1.35588 0.677942 0.735116i \(-0.262872\pi\)
0.677942 + 0.735116i \(0.262872\pi\)
\(558\) 0 0
\(559\) −8.00000 12.0000i −0.338364 0.507546i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.00000i 0.210912i
\(563\) 41.0000i 1.72794i 0.503540 + 0.863972i \(0.332031\pi\)
−0.503540 + 0.863972i \(0.667969\pi\)
\(564\) 7.00000i 0.294753i
\(565\) 0 0
\(566\) 6.00000i 0.252199i
\(567\) 2.00000 0.0839921
\(568\) 5.00000i 0.209795i
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 10.0000i 0.417392i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −13.0000 −0.540729
\(579\) 24.0000i 0.997406i
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 2.00000i 0.0829027i
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 32.0000 1.32078 0.660391 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 3.00000i 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000i 0.740421i
\(592\) 7.00000 0.287698
\(593\) 1.00000 0.0410651 0.0205325 0.999789i \(-0.493464\pi\)
0.0205325 + 0.999789i \(0.493464\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.0000i 0.573462i
\(597\) 15.0000i 0.613909i
\(598\) 8.00000 + 12.0000i 0.327144 + 0.490716i
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −23.0000 −0.938190 −0.469095 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 8.00000i 0.326056i
\(603\) −7.00000 −0.285062
\(604\) 0 0
\(605\) 0 0
\(606\) 2.00000i 0.0812444i
\(607\) 3.00000i 0.121766i 0.998145 + 0.0608831i \(0.0193917\pi\)
−0.998145 + 0.0608831i \(0.980608\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 10.0000i 0.405220i
\(610\) 0 0
\(611\) 21.0000 14.0000i 0.849569 0.566379i
\(612\) 2.00000i 0.0808452i
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −17.0000 −0.686064
\(615\) 0 0
\(616\) 0 0
\(617\) 7.00000 0.281809 0.140905 0.990023i \(-0.454999\pi\)
0.140905 + 0.990023i \(0.454999\pi\)
\(618\) 4.00000 0.160904
\(619\) 36.0000i 1.44696i 0.690344 + 0.723481i \(0.257459\pi\)
−0.690344 + 0.723481i \(0.742541\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 18.0000 0.721734
\(623\) 28.0000i 1.12180i
\(624\) −2.00000 3.00000i −0.0800641 0.120096i
\(625\) 0 0
\(626\) 11.0000i 0.439648i
\(627\) 0 0
\(628\) 22.0000i 0.877896i
\(629\) 14.0000i 0.558217i
\(630\) 0 0
\(631\) 50.0000i 1.99047i 0.0975126 + 0.995234i \(0.468911\pi\)
−0.0975126 + 0.995234i \(0.531089\pi\)
\(632\) 5.00000 0.198889
\(633\) 12.0000i 0.476957i
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −9.00000 −0.356873
\(637\) −9.00000 + 6.00000i −0.356593 + 0.237729i
\(638\) 0 0
\(639\) 5.00000i 0.197797i
\(640\) 0 0
\(641\) −28.0000 −1.10593 −0.552967 0.833203i \(-0.686504\pi\)
−0.552967 + 0.833203i \(0.686504\pi\)
\(642\) −3.00000 −0.118401
\(643\) −9.00000 −0.354925 −0.177463 0.984128i \(-0.556789\pi\)
−0.177463 + 0.984128i \(0.556789\pi\)
\(644\) 8.00000i 0.315244i
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) 38.0000i 1.49393i 0.664861 + 0.746967i \(0.268491\pi\)
−0.664861 + 0.746967i \(0.731509\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 46.0000i 1.80012i 0.435767 + 0.900060i \(0.356477\pi\)
−0.435767 + 0.900060i \(0.643523\pi\)
\(654\) −1.00000 −0.0391031
\(655\) 0 0
\(656\) 5.00000i 0.195217i
\(657\) 4.00000 0.156055
\(658\) −14.0000 −0.545777
\(659\) 5.00000 0.194772 0.0973862 0.995247i \(-0.468952\pi\)
0.0973862 + 0.995247i \(0.468952\pi\)
\(660\) 0 0
\(661\) 25.0000i 0.972387i 0.873851 + 0.486194i \(0.161615\pi\)
−0.873851 + 0.486194i \(0.838385\pi\)
\(662\) 20.0000i 0.777322i
\(663\) −6.00000 + 4.00000i −0.233021 + 0.155347i
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) 20.0000i 0.774403i
\(668\) 17.0000 0.657750
\(669\) 6.00000i 0.231973i
\(670\) 0 0
\(671\) 0 0
\(672\) 2.00000i 0.0771517i
\(673\) 11.0000i 0.424019i 0.977268 + 0.212009i \(0.0680008\pi\)
−0.977268 + 0.212009i \(0.931999\pi\)
\(674\) 22.0000i 0.847408i
\(675\) 0 0
\(676\) 5.00000 12.0000i 0.192308 0.461538i
\(677\) 22.0000i 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) −6.00000 −0.230429
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 12.0000i 0.459841i
\(682\) 0 0
\(683\) 46.0000 1.76014 0.880071 0.474843i \(-0.157495\pi\)
0.880071 + 0.474843i \(0.157495\pi\)
\(684\) 1.00000i 0.0382360i
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 21.0000 0.801200
\(688\) 4.00000i 0.152499i
\(689\) −18.0000 27.0000i −0.685745 1.02862i
\(690\) 0 0
\(691\) 5.00000i 0.190209i −0.995467 0.0951045i \(-0.969681\pi\)
0.995467 0.0951045i \(-0.0303185\pi\)
\(692\) 1.00000i 0.0380143i
\(693\) 0 0
\(694\) 33.0000i 1.25266i
\(695\) 0 0
\(696\) 5.00000i 0.189525i
\(697\) 10.0000 0.378777
\(698\) 6.00000i 0.227103i
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) −2.00000 3.00000i −0.0754851 0.113228i
\(703\) 7.00000i 0.264010i
\(704\) 0 0
\(705\) 0 0
\(706\) −1.00000 −0.0376355
\(707\) 4.00000 0.150435
\(708\) −4.00000 −0.150329
\(709\) 34.0000i 1.27690i −0.769665 0.638448i \(-0.779577\pi\)
0.769665 0.638448i \(-0.220423\pi\)
\(710\) 0 0
\(711\) 5.00000 0.187515
\(712\) 14.0000i 0.524672i
\(713\) 0 0
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 21.0000i 0.783713i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 8.00000i 0.297936i
\(722\) −18.0000 −0.669891
\(723\) −10.0000 −0.371904
\(724\) −8.00000 −0.297318
\(725\) 0 0
\(726\) 11.0000i 0.408248i
\(727\) 32.0000i 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) −6.00000 + 4.00000i −0.222375 + 0.148250i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 12.0000i 0.443533i
\(733\) 31.0000 1.14501 0.572506 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(734\) 13.0000i 0.479839i
\(735\) 0 0
\(736\) 4.00000i 0.147442i
\(737\) 0 0
\(738\) 5.00000i 0.184053i
\(739\) 11.0000i 0.404642i 0.979319 + 0.202321i \(0.0648484\pi\)
−0.979319 + 0.202321i \(0.935152\pi\)
\(740\) 0 0
\(741\) 3.00000 2.00000i 0.110208 0.0734718i
\(742\) 18.0000i 0.660801i
\(743\) −29.0000 −1.06391 −0.531953 0.846774i \(-0.678542\pi\)
−0.531953 + 0.846774i \(0.678542\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000i 0.951928i
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 6.00000i 0.219235i
\(750\) 0 0
\(751\) −3.00000 −0.109472 −0.0547358 0.998501i \(-0.517432\pi\)
−0.0547358 + 0.998501i \(0.517432\pi\)
\(752\) 7.00000 0.255264
\(753\) 13.0000i 0.473746i
\(754\) 15.0000 10.0000i 0.546268 0.364179i
\(755\) 0 0
\(756\) 2.00000i 0.0727393i
\(757\) 12.0000i 0.436147i −0.975932 0.218074i \(-0.930023\pi\)
0.975932 0.218074i \(-0.0699773\pi\)
\(758\) 16.0000i 0.581146i
\(759\) 0 0
\(760\) 0 0
\(761\) 45.0000i 1.63125i −0.578582 0.815624i \(-0.696394\pi\)
0.578582 0.815624i \(-0.303606\pi\)
\(762\) 17.0000 0.615845
\(763\) 2.00000i 0.0724049i
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −21.0000 −0.758761
\(767\) −8.00000 12.0000i −0.288863 0.433295i
\(768\) 1.00000i 0.0360844i
\(769\) 4.00000i 0.144244i −0.997396 0.0721218i \(-0.977023\pi\)
0.997396 0.0721218i \(-0.0229770\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) −24.0000 −0.863779
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 4.00000i 0.143777i
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 14.0000i 0.502247i
\(778\) −25.0000 −0.896293
\(779\) −5.00000 −0.179144
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 5.00000i 0.178685i
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 13.0000i 0.463695i
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) −18.0000 −0.641223
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 12.0000i 0.426671i
\(792\) 0 0
\(793\) 36.0000 24.0000i 1.27840 0.852265i
\(794\) 33.0000 1.17113
\(795\) 0 0
\(796\) 15.0000 0.531661
\(797\) 42.0000i 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 14.0000i 0.495284i
\(800\) 0 0
\(801\) 14.0000i 0.494666i
\(802\) 10.0000i 0.353112i
\(803\) 0 0
\(804\) 7.00000i 0.246871i
\(805\) 0 0
\(806\) 0 0
\(807\) 15.0000i 0.528025i
\(808\) −2.00000 −0.0703598
\(809\) 40.0000 1.40633 0.703163 0.711029i \(-0.251771\pi\)
0.703163 + 0.711029i \(0.251771\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i −0.936320 0.351147i \(-0.885792\pi\)
0.936320 0.351147i \(-0.114208\pi\)
\(812\) −10.0000 −0.350931
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 4.00000 0.139942
\(818\) 16.0000i 0.559427i
\(819\) −6.00000 + 4.00000i −0.209657 + 0.139771i
\(820\) 0 0
\(821\) 40.0000i 1.39601i 0.716093 + 0.698005i \(0.245929\pi\)
−0.716093 + 0.698005i \(0.754071\pi\)
\(822\) 13.0000i 0.453427i
\(823\) 49.0000i 1.70803i −0.520246 0.854016i \(-0.674160\pi\)
0.520246 0.854016i \(-0.325840\pi\)
\(824\) 4.00000i 0.139347i
\(825\) 0 0
\(826\) 8.00000i 0.278356i
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 3.00000 2.00000i 0.104006 0.0693375i
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −15.0000 −0.518166
\(839\) 16.0000i 0.552381i 0.961103 + 0.276191i \(0.0890721\pi\)
−0.961103 + 0.276191i \(0.910928\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 30.0000i 1.03387i
\(843\) 5.00000 0.172209
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 7.00000 0.240665
\(847\) 22.0000 0.755929
\(848\) 9.00000i 0.309061i
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) 28.0000i 0.959828i
\(852\) 5.00000 0.171297
\(853\) 31.0000 1.06142 0.530710 0.847554i \(-0.321925\pi\)
0.530710 + 0.847554i \(0.321925\pi\)
\(854\) −24.0000 −0.821263
\(855\) 0 0
\(856\) 3.00000i 0.102538i
\(857\) 32.0000i 1.09310i −0.837427 0.546550i \(-0.815941\pi\)
0.837427 0.546550i \(-0.184059\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 15.0000i 0.510902i
\(863\) 51.0000 1.73606 0.868030 0.496512i \(-0.165386\pi\)
0.868030 + 0.496512i \(0.165386\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 0 0
\(866\) 19.0000i 0.645646i
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 21.0000 14.0000i 0.711558 0.474372i
\(872\) 1.00000i 0.0338643i
\(873\) −2.00000 −0.0676897
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 4.00000i 0.135147i
\(877\) −53.0000 −1.78968 −0.894841 0.446384i \(-0.852711\pi\)
−0.894841 + 0.446384i \(0.852711\pi\)
\(878\) 5.00000 0.168742
\(879\) 6.00000i 0.202375i
\(880\) 0 0
\(881\) 52.0000 1.75192 0.875962 0.482380i \(-0.160227\pi\)
0.875962 + 0.482380i \(0.160227\pi\)
\(882\) −3.00000 −0.101015
\(883\) 34.0000i 1.14419i −0.820187 0.572096i \(-0.806131\pi\)
0.820187 0.572096i \(-0.193869\pi\)
\(884\) −4.00000 6.00000i −0.134535 0.201802i
\(885\) 0 0
\(886\) 21.0000i 0.705509i
\(887\) 32.0000i 1.07445i −0.843437 0.537227i \(-0.819472\pi\)
0.843437 0.537227i \(-0.180528\pi\)
\(888\) 7.00000i 0.234905i
\(889\) 34.0000i 1.14032i
\(890\) 0 0
\(891\) 0 0
\(892\) 6.00000 0.200895
\(893\) 7.00000i 0.234246i
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −12.0000 + 8.00000i −0.400668 + 0.267112i
\(898\) 9.00000i 0.300334i
\(899\) 0 0
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 6.00000i 0.199557i
\(905\) 0 0
\(906\) 0 0
\(907\) 2.00000i 0.0664089i −0.999449 0.0332045i \(-0.989429\pi\)
0.999449 0.0332045i \(-0.0105712\pi\)
\(908\) 12.0000 0.398234
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 21.0000i 0.693860i
\(917\) −26.0000 −0.858596
\(918\) −2.00000 −0.0660098
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) 0 0
\(921\) 17.0000i 0.560169i
\(922\) 0 0
\(923\) 10.0000 + 15.0000i 0.329154 + 0.493731i
\(924\) 0 0
\(925\) 0 0
\(926\) 34.0000 1.11731
\(927\) 4.00000i 0.131377i
\(928\) 5.00000 0.164133
\(929\) 29.0000i 0.951459i −0.879592 0.475730i \(-0.842184\pi\)
0.879592 0.475730i \(-0.157816\pi\)
\(930\) 0 0
\(931\) 3.00000i 0.0983210i
\(932\) 24.0000i 0.786146i
\(933\) 18.0000i 0.589294i
\(934\) 23.0000i 0.752583i
\(935\) 0 0
\(936\) 3.00000 2.00000i 0.0980581 0.0653720i
\(937\) 22.0000i 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) −14.0000 −0.457116
\(939\) 11.0000 0.358971
\(940\) 0 0
\(941\) 30.0000i 0.977972i 0.872292 + 0.488986i \(0.162633\pi\)
−0.872292 + 0.488986i \(0.837367\pi\)
\(942\) 22.0000 0.716799
\(943\) 20.0000 0.651290
\(944\) 4.00000i 0.130189i
\(945\) 0 0
\(946\) 0 0
\(947\) 42.0000 1.36482 0.682408 0.730971i \(-0.260933\pi\)
0.682408 + 0.730971i \(0.260933\pi\)
\(948\) 5.00000i 0.162392i
\(949\) −12.0000 + 8.00000i −0.389536 + 0.259691i
\(950\) 0 0
\(951\) 18.0000i 0.583690i
\(952\) 4.00000i 0.129641i
\(953\) 14.0000i 0.453504i −0.973952 0.226752i \(-0.927189\pi\)
0.973952 0.226752i \(-0.0728108\pi\)
\(954\) 9.00000i 0.291386i
\(955\) 0 0
\(956\) 24.0000i 0.776215i
\(957\) 0 0
\(958\) 31.0000i 1.00156i
\(959\) −26.0000 −0.839584
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) −21.0000 + 14.0000i −0.677067 + 0.451378i
\(963\) 3.00000i 0.0966736i
\(964\) 10.0000i 0.322078i
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) −58.0000 −1.86515 −0.932577 0.360971i \(-0.882445\pi\)
−0.932577 + 0.360971i \(0.882445\pi\)
\(968\) −11.0000 −0.353553
\(969\) 2.00000i 0.0642493i
\(970\) 0 0
\(971\) 17.0000 0.545556 0.272778 0.962077i \(-0.412058\pi\)
0.272778 + 0.962077i \(0.412058\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 4.00000i 0.127906i
\(979\) 0 0
\(980\) 0 0
\(981\) 1.00000i 0.0319275i
\(982\) −12.0000 −0.382935
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) −5.00000 −0.159394
\(985\) 0 0
\(986\) 10.0000i 0.318465i
\(987\) 14.0000i 0.445625i
\(988\) 2.00000 + 3.00000i 0.0636285 + 0.0954427i
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 10.0000i 0.317181i
\(995\) 0 0
\(996\) 4.00000i 0.126745i
\(997\) 48.0000i 1.52018i 0.649821 + 0.760088i \(0.274844\pi\)
−0.649821 + 0.760088i \(0.725156\pi\)
\(998\) 31.0000i 0.981288i
\(999\) 7.00000i 0.221470i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1950.2.f.e.649.1 2
5.2 odd 4 1950.2.b.a.1351.1 2
5.3 odd 4 1950.2.b.f.1351.2 yes 2
5.4 even 2 1950.2.f.f.649.2 2
13.12 even 2 1950.2.f.f.649.1 2
65.12 odd 4 1950.2.b.a.1351.2 yes 2
65.38 odd 4 1950.2.b.f.1351.1 yes 2
65.64 even 2 inner 1950.2.f.e.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.b.a.1351.1 2 5.2 odd 4
1950.2.b.a.1351.2 yes 2 65.12 odd 4
1950.2.b.f.1351.1 yes 2 65.38 odd 4
1950.2.b.f.1351.2 yes 2 5.3 odd 4
1950.2.f.e.649.1 2 1.1 even 1 trivial
1950.2.f.e.649.2 2 65.64 even 2 inner
1950.2.f.f.649.1 2 13.12 even 2
1950.2.f.f.649.2 2 5.4 even 2